[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123366-en":3,"doc-seo-123366-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},123366,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Exploring Representations and Inductive Bias for Machine Learning Tasks in Knot Theory","Knot theory studies circle embeddings in R3 under ambient isotopy, where knots and links carry invariants such as integer-valued functions or polynomial families. The thesis investigates predicting such invariants to better understand knot structure and to enable useful machine learning representations. Supervised experiments use five models with three representations on a large dataset, introducing a knot- and link-to-graph conversion for graph neural networks, which outperforms baselines.","Brigham Young University  \nBYU ScholarsArchive  \nTheses and Dissertations  \n2025-04-23  \nExploring Representations and Inductive Bias for Machine Learning Tasks in Knot Theory  \nNathaniel Driggs  \nBrigham Young University  \nFollow this and additional works at: [https://scholarsarchive.byu.edu/etd](https://scholarsarchive.byu.edu/etd)  \n Part of the Physical Sciences and Mathematics Commons  \nBYU ScholarsArchive Citation  \nDriggs, Nathaniel, \"Exploring Representations and Inductive Bias for Machine Learning Tasks in Knot Theory\" (2025) . Theses and Dissertations. 10813.  \n[https://scholarsarchive.byu.edu/etd/10813](https://scholarsarchive.byu.edu/etd/10813)  \nThis Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please [contact ellen_amatangelo@byu.edu](contact ellen_amatangelo@byu.edu).  \nExploring Representations and Inductive Bias for Machine Learning Tasks in Knot Theory  \nNathaniel Driggs  \nA thesis submitted to the faculty of  \nBrigham Young University  \nin partial fulfillment of the requirements for the degree of  \nMaster of Science  \nMark Hughes, Chair  \nDan Ventura  \nJared Whitehead  \nDepartment of Mathematics  \nBrigham Young University  \nCopyright © 2025 Nathaniel Driggs  \nAll Rights Reserved  \nabstract  \nExploring Representations and Inductive Bias for Machine Learning Tasks in Knot Theory  \nNathaniel Driggs  \nDepartment of Mathematics, BYU  \nMaster of Science  \nKnot theory is a branch of mathematics that studies embeddings of the circle in R3 that are equivalent up to ambient isotopy. A link is a knot with one or more component. Knots and links have invariants, or functions from each knot to a set such as the integers or a family of polynomials. Predicting invariants can help us approximate them, understand them better, and find useful ways of representing knots and useful models for other machine learning tasks in knot theory.  \nI perform many supervised learning experiments, predicting signature on a large, wideranging dataset using 5 di↵erent models and 3 di↵erent representations. We present a new method of converting knots and links to graph data, prior to being fed into a graph neural network (GNN) . The GNN outperforms the other models, unveiling a new tool for machine learning tasks in knot theory.  \nAdditionally, we attempted to use reinforcement learning to find candidate knots to test for disproving a conjecture concerning an invariant called the Jones polynomial. We present a reinforcement learning environment for building links to maximize or minimize di↵erent invariants. Despite successfully maximizing and minimizing the desired invariants, the agent always generates links not knots, so no candidates . We re produced. We suggest future directions for finding candidate knots and applying GNNs to new tasks. The code for this thesis can be found at [https://github.com/ndriggs/conditional-link-generation](https://github.com/ndriggs/conditional-link-generation).  \nKeywords: knot theory, supervised learning, graph neural networks, reinforcement learning  \nAcknowledgements  \nThank you to my advisor Mark Hughes for giving me free reign to explore my own knot theory machine learning ideas, and supporting me in my endeavors. Thank you to my parents for being incredibly loving and supportive through good times and bad, I’m so glad we was born to them. Thanks to my siblings for their support, we feel so lucky to have gotten to spend the 2 years of this masters here in Provo with 3 of them. And thank you tomy roommates of these past two years, you make living fun and enjoyable.  \nContents  \nContents iv  \nList of Tables vi  \nList of Figures vii  \n1 Knot Theory 1  \n1.1 Knot diagrams .................................. 1  \n1.2 Braids ....................................... 2  \n1.3 Invariants ..................................... 2  \n1.4 Problem Motivation ......","cbCaig3cG6R8oSzR","https://ap.wps.com/l/cbCaig3cG6R8oSzR","pdf",1538126,1,50,"English","en",105,"# Knot Theory\n## Knot diagrams\n## Braids\n## Invariants\n## Problem Motivation\n# Supervised Learning\n## Predicting Invariants\n## Generating the Dataset\n## Lawrence-Krammer Representation\n## Multi-Layer Perceptron\n## Convolutional Neural Network\n## Transformers\n## Graph Neural Networks\n## Implementation\n## Results\n# Reinforcement Learning\n## Background\n## Link Builder Environment\n## State Representations\n## Results\n# Future Work\n## Reinforcement Learning\n## Generating Knots\n## Supervised Learning\n## Variational Autoencoder (VAE)\n## Large Language Models\n## Conclusion","[{\"question\":\"What is the main goal of the thesis in machine learning for knot theory?\",\"answer\":\"The thesis aims to improve machine learning tasks in knot theory by studying representations and inductive bias, focusing on predicting knot and link invariants.\"},{\"question\":\"How are knots and links represented for graph neural networks?\",\"answer\":\"The work presents a method to convert knots and links into graph data, which is then fed into a graph neural network for learning and prediction.\"},{\"question\":\"Why did the reinforcement learning approach fail to produce knot candidates?\",\"answer\":\"Although the environment can maximize or minimize target invariants, the agent consistently generates links rather than knots, preventing valid candidate knots for the conjecture test.\"}]","Exploring Representations and Inductive Bias for Machine Learning Tasks in Knot Theory | 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